# If license plates have three numbers and three letters in any order, how many unique license plates can there be?

Oct 13, 2015

$351520000$

#### Explanation:

There are 10 possible digits for the numbers (0,1,2,...,9), and 26 possible letters (A,B,C,.....,Z).

Since repetitions are allowed, we have that for each letter used, 26 still remain for the next choice, and for each digit used, 10 still remain for the next choice.

Hence, by the multiplication principle, the total different number of letter permutations is $26 \times 26 \times 26 = 17576$ and the total number of digit permutations is $10 \times 10 \times 10 = 1000$

But now since they can be in any order, we basically require in how many different ways can 3 be chosen from 6 which is the combination
""^6C_3=(6!)/((6-3)!3!)=20

So for each of the original permutations of numbers and letters, we must consider the 20 different combinations of positions they can be in.

Therefore by the multiplication principle, the total possible different number plates is
$17576 \times 1000 \times 20 = 351520000$